Chemistry Reference
In-Depth Information
Since the time dependence of observables in the course of evolution of the
density matrix is described by more than one exponential, one needs an appropriate
definition of the relaxation rate or relaxation time. A convenient way is to use the
integral relaxation time defined as the area under the relaxation curve
0
dt
[
A
(
t
)
−
A
(
∞
)]
τ
int
≡
(94)
A
(0)
−
A
(
∞
)
One can check that in the case of a single exponential,
)]
e
−
t
A
(
t
)
=
A
(
∞
)
+
[
A
(0)
−
A
(
∞
(95)
the result is
τ
int
=
1
/.
From Eq. (92), one obtains
A
R
μ
e
−
μ
t
L
μ
·
ρ
(0)
A
(
t
)
−
A
(
∞
)
=
·
(96)
μ/
=
1
and thus
2
A
R
μ
−
μ
L
μ
·
ρ
(0)
N
2
μ
·
=
τ
int
=
(97)
2
A
R
μ
L
μ
·
ρ
(0)
N
2
μ
·
=
This formula cannot be fully vectorized since summation skips the static eigenvalue
μ
=
1
.
2.
Secular DME
Within the secular approximation, one considers the dynamics of diagonal and
nondiagonal components of the density matrix separately. The former is described
by Eqs. (80)-(86), where the vector
ρ
is replaced by the vector of the diagonal
and the
N
2
N
2
matrix
components
n
={
n
α
}={
ρ
αα
}
×
is replaced by the
N
×
sec
having matrix elements
N
matrix
δ
αα
γ
sec
αα
=
(1
−
δ
αα
)
αα
−
γα
(98)
sec
are positive reals, except
as follows from Eqs. (63) or (67). All eigenvalues of
for one zero eigenvalue,
1
=
0
.
Equation (87) becomes simply
exp
1
Z
s
ε
α
k
B
T
L
1
α
=
1
,
1
α
=
−
(99)
If the initial condition is a diagonal matrix, the nondiagonal elements do not arise
dynamically, and hence they can be dropped. Then the time dependence of any
